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Conditions for Bar Formation in Bulgeless Disk Galaxies

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Two numbers predict which bulgeless disk galaxies grow bars.

desk verdict A clean simulation suite and a sensible two-parameter bar criterion, but the Γ=10 boundary is a fit to the same data, so verify out-of-sample. read the letter →

arxiv 2509.07353 v1 pith:2I356L5I submitted 2025-09-09 astro-ph.GA

Dajeong Jang (SNU) , Woong-Tae Kim (SNU) , Yun Hee Lee (KNU) This is my paper · ORCID
classification astro-ph.GA
keywords galaxybarsbulgelessdiskgalaxiesswingamplificationToomrestabilityparameterN-bodysimulationsbarformationcriterionbucklinginstabilityS4Gsurvey
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to pin down when a disk galaxy without a classical bulge becomes unstable to bar formation. Using N-body simulations of 23 models with stellar masses from 10^9 to 10^11 solar masses, the authors show that the outcome is governed by two local disk parameters, the Toomre stability parameter Q_T and the dimensionless wavelength X, through the swing amplification factor Gamma. They propose the criterion Q_T,bar + 0.4(X_bar - 1.4)^2 <= 1.8, equivalent to Gamma >= 10: every bar-forming model in the suite satisfies it, and every stable model avoids it. The paper also reports that bars in low-mass galaxies are short, weak, and easily disrupted by outer spiral arms, while bars in high-mass galaxies are long, strong, and prone to vertical buckling. The claim matters because existing one-parameter criteria such as the Ostriker-Peebles and ELN conditions fail to separate the bar-forming from the stable models in these simulations.

What carries the argument

The load-bearing object is the swing amplification factor Gamma of a local razor-thin infinite disk, defined as the peak perturbed displacement of a shearing wavelet divided by its initial displacement. From the standard swing-amplification differential equation, Gamma depends only on Q_T and X; the paper evaluates it for each galaxy using values averaged over the inner disk (2 kpc to the radius of minimum Q_T). The criterion Q_T,bar + 0.4(X_bar - 1.4)^2 <= 1.8 is the Gamma = 10 contour in this two-parameter plane, which separates bar-forming and stable models in the simulations.

What would settle it

A bulgeless disk simulation designed to have radially averaged Q_T,bar = 1.5 and X_bar = 3.0, which the criterion predicts lies below the Gamma = 10 contour and should stay bar-free, would falsify the criterion if a bar nonetheless forms within 10 Gyr; equivalently, measuring Q_T and X in real low-mass galaxies and finding that many barred galaxies sit below the Gamma = 10 contour would show the criterion is not the separator.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that bar formation in bulgeless disk galaxies is controlled by the efficiency of swing amplification, a local shearing-disk process in which leading spiral perturbations grow as they wind into trailing waves. When the radially averaged Toomre parameter Q_T,bar and dimensionless azimuthal wavelength X_bar, averaged over 2 kpc <= R <= R_QT,min, fall in the region where the amplification factor Gamma reaches or exceeds 10, the disk develops a bar within 10 Gyr; otherwise it stays stable. This boundary is approximately Q_T,bar + 0.4(X_bar - 1.4)^2 <= 1.8, which the authors stress is preferable to one-parameter criteria because Q_T and X act inde

Load-bearing premise

The criterion rests on the assumption that a local, razor-thin, infinite-disk swing amplification calculation, with shear parameter q = 1 and radial averaging from 2 kpc to R_QT,min, captures the global bar instability of the finite-thickness 3D simulated disks.

Editorial extensions

If this is right

  • For bulgeless galaxy models, bar formation can be read off directly from two easily computed disk quantities without running a simulation to 10 Gyr.
  • Traditional one-parameter stability indicators, t_OP and epsilon_ELN, do not separate bar-forming from stable bulgeless disks in this mass range.
  • Low-mass disks, when they do form bars, produce short weak bars that can be destroyed by outer spiral arms, while high-mass disks form long strong bars that survive and undergo buckling.
  • Buckling instability is not triggered by low sigma_z/sigma_R alone: low-mass bars remain vertically thin even with sigma_z/sigma_R below 0.55.
  • Observed trends of bar strength and length increasing with stellar mass are reproduced, with simulated bars somewhat stronger and about 60 percent longer, plausibly because the models lack a classical bulge.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The criterion is formulated for isolated bulgeless systems; a natural test is whether adding gas shifts the Gamma = 10 boundary, since gas both cools the disk and changes the effective surface density and velocity dispersion.
  • If the local-to-global mapping holds, observed galaxies with measured rotation curves and velocity dispersions could be placed in the Q_T-X plane to predict their bar-forming likelihood, giving a direct observational check of the criterion.
  • The threshold Gamma = 10 is calibrated on a 10 Gyr window and on a particular halo density profile; galaxies with different halo concentration or longer evolution times could form bars just below this boundary, so the inequality may be a practical rather than absolute threshold.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper uses 23 collisionless N-body simulations of isolated, bulgeless disk galaxies spanning stellar masses 10^9–10^11 M_sun, grouped into four mass bins, with halo scale radius varied within each group. It reports that bars form through repeated swing amplification with feedback, and proposes a two-parameter bar-formation criterion, Q_T,bar + 0.4(X_bar - 1.4)^2 <= 1.8 (Eq. 17), corresponding to a swing-amplification factor Gamma >= 10. The criterion is evaluated using radially averaged initial Q_T and X over 2 kpc <= R <= R_QT,min (Table 1). The paper also presents mass-dependent trends in bar length, strength, pattern speed, spiral interaction, and buckling instability, comparing these with observations.

Significance. The study is potentially valuable: it provides a systematic, high-resolution simulation suite over a broad galaxy mass range, uses observationally motivated initial conditions from S4G, and carefully separates bar-forming from stable cases. The comparison with traditional one-parameter criteria (t_OP and epsilon_ELN) is useful, and the mass-dependent bar properties and buckling behavior are of independent interest. The central claim, however, is that Eq. (17) is a general condition for bar formation in bulgeless disks. That claim is presently supported only by the same 23 simulations from which the threshold, radial averaging range, and quadratic coefficients were calibrated. If Eq. (17) were validated on independent initial conditions or with robustness tests, it would be a significant advance; as it stands, it is an interesting empirical separator rather than an established predictive criterion.

major comments (3)
  1. [Section 4.1, Eqs. (13)-(16), and Figure 2] The proposed criterion is calibrated in-sample. The Gamma = 10 boundary is identified after the fact from the same simulation outcomes it is then claimed to 'account for', and the coefficients 0.4 and 1.8 in Eq. (17) are fits to that boundary. The radial averaging range (2 kpc <= R <= R_QT,min) is also chosen by hand, without a sensitivity study. Since every free element of the criterion is set using these 23 models, the clean separation in Figure 16 does not demonstrate predictive power. I recommend either explicitly reframing Eq. (17) as an empirical calibration and adding out-of-sample tests, or adding robustness tests such as different particle noise realizations, different halo profiles, and variation of the averaging range to show that the boundary and its location are stable.
  2. [Section 3.1, Table 1, Figure 5] The analytic amplification calculation assumes a razor-thin, infinite disk with shear parameter q = 1 and initial kx(0) = 0. The actual models are 3D and have finite thickness, as stated in Section 2.2. More importantly, the rotation curves shown in Figure 2 are not flat in the bar-forming region, especially Groups 1 and 2 where v_rot is still rising over 2 kpc <= R <= R_QT,min; q = 1 is therefore not representative of these disks. The local-to-global mapping of a single-amplification calculation onto a global bar instability also needs justification. Concretely, the authors should quantify how Gamma changes with the actual local shear q(R) and with the vertical thickness (e.g., a reduced surface density or finite-thickness reduction factor), and show that the Gamma >= 10 boundary and Eq. (17) are robust to these variations.
  3. The separation between bar-forming and stable models is narrow in several cases. In Table 1, stable G1A31 has (Q_T,bar, X_bar) = (1.5, 2.4) while bar-forming G1A34 has (1.4, 2.2); similarly G2A28 (1.5, 2.3) is stable while G2A30 (1.4, 2.1) forms a bar. G4A45, which does form a bar, lies essentially on the Eq. (17) boundary. With only 23 models, no multiple realizations, and a fixed 10 Gyr integration time, the apparent threshold may depend on the A2/A0 >= 0.2 bar criterion and on whether a slowly growing bar has had enough time to emerge. I ask for convergence tests, multiple noise realizations for at least the boundary models, and/or longer integrations to confirm that the Gamma >= 10 boundary is not an artifact of finite runtime or stochastic initial conditions.
minor comments (5)
  1. [Section 3.2, Fig. 20] The caption and text refer to 'COO' where the model is named C00 elsewhere; please correct the typo.
  2. [Table 1, Section 3.1] The symbol R is used for radius, for the ratio R_CR/R_bar, and for the corotation radius in Figure 20. This overloading is confusing; please use a distinct symbol for the ratio (e.g., R_CR/R_bar or script R).
  3. [Section 4.1, Eq. (16)] The bar formation time t_bar is listed but its precise definition is not given. Is it the first time A2/A0 >= 0.2, or the time when the bar length/pattern-speed criteria are met? Please state the measurement rule.
  4. [Section 2.2, Table 1 note] Equation (16) defines F(ν, x) with ν^2 = S(t)/kappa_0^2, while S(t) itself depends on F through Eq. (14). Please state how this implicit equation is solved in the integration and whether iteration is used.
  5. The footnote in Table 1 and the text explain that the range of a_h is not intended to match observed V_max values but to span stable and unstable models. This is honest and useful, but it should be stated more prominently in Section 5.1 where the conclusions about low-mass galaxies are drawn, to avoid the impression that the simulated sample reproduces the observed V_max distribution.

Circularity Check

1 steps flagged · score 6.0 of 10

The Γ=10 threshold in Eq. (17) is calibrated to the same simulations it is then said to predict; the criterion's boundary is an in-sample classifier rather than an independently derived bar-formation condition.

  1. fitted input called prediction [Section 4.2, Eq. (17) and Figure 16; reiterated in Section 5.1]
    "Note that all bar-forming models fall within the region characterized by an amplification factor of Γ≳10. In contrast, models in the region with Γ<10 undergo swing amplification that, even when sustained by repeated feedback loops, remains insufficient to trigger bar formation within 10 Gyr. The region with Γ≳10 is well approximated by QT,bar + 0.4(Xbar −1.4)^2 ≤1.8"

    The analytic computation of Γ(QT,X) from the local swing-amplification theory is independent and gives the shape of the amplification surface, but the threshold Γ=10 is not predicted by that theory. It is selected post hoc so that all 23 of the authors' own bar-forming models lie on one side and the stable models on the other. Equation (17) is then the analytic approximation to this chosen contour. The paper presents Eq. (17) as a 'criterion' and 'shows' that bar-forming models satisfy it, but this is an in-sample separation: the level Γ=10 was chosen to achieve exactly that separation. No out-of-sample test or independent calibration is provided. The additional choice of the radial averaging range 2 kpc ≤ R ≤ R_QT,min is also made without sensitivity analysis, so the resulting boundary is

full rationale

The paper contains substantial independent content: the N-body simulations, the bar-detection criterion, the evolution of bar properties, comparisons with observations, and the buckling analysis are all self-contained and do not reduce to the paper's own inputs. The swing-amplification calculation in Section 4.1 is standard theory and is not circular. However, the central bar-formation criterion of Eq. (17) is calibrated to the same simulations it is then used to 'predict'. The theoretical Γ surface gives the functional form of the boundary, but the level Γ=10 is a free parameter fixed by the authors' simulation outcomes. Consequently, the statement that all bar-forming models satisfy Γ≳10 is a restatement of the calibration choice, not an independent verification. The averaging range over which QT,bar and Xbar are computed is also chosen without independent justification. These features make the central claim partially circular: the criterion is an in-sample classifier. The paper is transparent about proposing this criterion based on the simulations, which prevents a score of 8 or 10, but the lack of an independent determination of the threshold and the absence of out-of-sample validation justify a score of 6. Self-citations to Jang & Kim (2023, 2024) are used for comparison and context, but they are not load-bearing for the new criterion, so they do not raise the circularity score further.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central criterion rests on a standard analytic amplification theory plus several modeling choices. The most consequential are the Gamma = 10 threshold and the radial averaging range, both calibrated to the simulations. No new physical entities are introduced. The local swing amplification model's applicability to finite-thickness disks is the main domain assumption.

free parameters (3)
  • Gamma threshold for bar formation = 10
    The boundary Gamma = 10 is chosen post hoc to separate bar-forming from stable models in Figure 16. It is not derived from first principles or independently predicted.
  • Radial averaging range for Q_T,bar and X_bar = 2 kpc <= R <= R_QT,min
    The averaging interval is chosen by hand to represent the bar-forming region. A different choice would shift the fitted boundary.
  • Coefficients in Equation (17) = 0.4 and 1.8
    These numbers are fit to the simulation outcomes to approximate the Gamma = 10 contour in the Q_T,bar-X_bar plane.
assumptions (3)
  • domain assumption Local swing amplification theory (Toomre 1981) with q = 1, razor-thin infinite disk, and initial condition kx(0) = 0, xi_dot = 0 at tau = -30
    Section 4.1 uses this local model to compute Gamma for each galaxy. Real disks are thick, finite, and have radially varying Q_T and X; the local-to-global mapping is assumed, not proven.
  • domain assumption Bar formation criterion A2/A0 >= 0.2 identifies a bar
    Section 3.1 adopts Algorry et al. (2017) threshold. Changing this threshold would change which models are classified as bar-forming and hence the fitted boundary.
  • domain assumption The feedback loop sustaining repeated swing amplification is sufficient for bar formation in the simulated disks
    Section 3.1 argues that trailing waves reflect or nonlinearly regenerate leading waves, but the paper does not derive the conditions under which this loop persists across mass scales.

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Cite this review

Pith. "Pith review of Conditions for Bar Formation in Bulgeless Disk Galaxies." pith.science (2026). https://pith.science/paper/2I356L5I

@misc{pith2026250907353,
  author       = {Pith},
  title        = {Pith review of: Conditions for Bar Formation in Bulgeless Disk Galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2I356L5I}},
  note         = {Machine review of arXiv:2509.07353}
}
abstract

While bars are commonly observed in disk galaxies, the precise conditions governing their formation remain incompletely understood. To investigate these conditions, we perform a suite of N-body simulations of bulgeless disk galaxies with stellar masses in the range $10^9 \leq M_d \leq 10^{11} \;M_\odot$. Our galaxy models are constructed based on the observed properties of nearby barred galaxies from the S4G survey, and we systematically vary the halo scale radius to isolate its dynamical influence. Bars in our simulations form via repeated swing amplifications of disk perturbations, sustained by feedback loops. The amplification factor $\Gamma$ depends on both the Toomre stability parameter $Q_T$ and the dimensionless wavelength $X$. Based on our simulation results, we propose a two-parameter bar formation criterion, $Q_T + 0.4(X - 1.4)^2 \leq 1.8$, corresponding to $\Gamma = 10$, which better captures the onset of bar formation than traditional one-parameter conditions. Bars in low-mass galaxies tend to be shorter and weaker, and are more susceptible to disruption by outer spiral arms. In contrast, bars in high-mass galaxies are longer, stronger, and more resilient to spiral interference. Bars in low-mass galaxies undergo only slight vertical thickening over time, whereas those in high-mass galaxies thicken rapidly via buckling instability.

Figures

Figures reproduced from arXiv: 2509.07353 by the authors.

Figure 1
Figure 1. Distribution of S4G barred galaxies from D´ıaz￾Garc´ıa et al. (2016) in the plane defined by the maximum H I circular velocity, Vmax, and stellar mass, M∗. The top and right panels draw the histograms of M∗ and Vmax, respec￾tively. The blue star symbol marks the location of the Milky Way, while the red squares with error bars indicate the mean and standard deviation within each of the four stellar mass bins. M∗/L∗ =… view at source ↗
Figure 2
Figure 2. Initial distributions of the circular velocity vrot as a function of radius, shown as solid lines for the models in (a) Group 1, (b) Group 2, (c) Group 3, and (d) Group 4. The dashed line in each panel indicates the contribution from the disk component. where r = (R2 + z 2 ) 1/2 is the spherical radius, and Mh and ah are the mass and the scale radius of the halo. Using the halo-to-stellar mass relation from D´ıaz￾Ga… view at source ↗
Figure 3
Figure 3. Radial distributions of (a) the Toomre stability parameter QT , (b) the dimensionless azimuthal wavelength X, and (c) frequencies Ω (solid) and Ω − κ/2 (dashed) at t = 0. In (a), a small dot marks the location of the minimum QT for each model. models. Overall, QT reaches a minimum at R ∼ (1.1– 1.4)Rd, increasing toward smaller radii as κ rises and toward larger radii as Σ declines. Note that X in￾creases monotonical… view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Left: Evolution of the maximum wavenumber pmax that maximizes the Fourier amplitude |A(2, p)|, as defined in Equation (7), for the m = 2 logarithmic spirals in model G3A29, along with the corresponding maximum amplitude Amax = |A(2, pmax)|. The shaded regions labeled (…
Figure 5
Figure 5. Figure 5: Temporal changes of the bar strength A2/A0 for all models in (a) Group 1, (b) Group 2, (c) Group 3, and (d) Group 4. The shaded vertical bands in (c) and (d) indicate the periods of buckling instability. t ∼ 6–7 Gyr. At later times, however, the spirals evolve differen…
Figure 6
Figure 6. Figure 6: Logarithm of the disk surface density Σ at selected times for model G1A40, G2A36, G3A35 and G4A60, from left to right. The time, t, is given in Gyr and the colorbars at the top label log Σ/(M⊙ pc−2 ). ing weaker and shorter, a process discussed further in Section 3.3. …
Figure 7
Figure 7. Figure 7: Contours of the normalized cross-correlation of the perturbed surface density in the radius-frequency plane for the snapshots shown in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Temporal evolution of the bar length Rbar (left) and RCR (right) for the bar-forming models in Group 1, Group 2, Group 3, and Group 4 from top to bottom. The thin portions of the curves for models G1A37, G1A40, G2A32, G2A34, and G2A36 indicate phases during which the b…
Figure 9
Figure 9. Figure 9: Temporal evolution of the bar pattern speed Ωb for the bar-forming models in (a) Group 1, (b) Group 2, (c) Group 3, and (d) Group 4. The thin portions of the curves for models G1A37, G1A40, G2A32, G2A34, and G2A36 indicate phases during which the bars are disrupted by …
Figure 10
Figure 10. Figure 10: Temporal evolution of the BPS strength Ps for all bar-forming models in (a) Group 1, (b) Group 2, (c) Group 3, and (d) Group 4. In (c) and (d), the shaded vertical bands indicate the periods of buckling instability. 10 5 0 5 10 4 2 0 2 4 z ( k p c ) G1A40 10 5 0 5 10 …
Figure 11
Figure 11. Figure 11: Contours of the logarithm of the projected disk densities at the end of the runs (t = 10 Gyr) for the fiducial models. The x- and z-axes correspond to the semimajor axis of the bar and the vertical direction, respectively. Dotted contours represent projected densities…
Figure 12
Figure 12. Figure 12: Temporal evolution of the mean vertical velocity, |⟨vz⟩|, measured in an annulus centered at R = 3 kpc for all bar-forming models in (a) Group 1, (b) Group 2, (c) Group 3, and (d) Group 4. In (c) and (d), the shaded vertical bands indicate the duration of the buckling…
Figure 13
Figure 13. Figure 13: Temporal evolution of the ratio σz/σR of the vertical to radial velocity dispersions of the disk particles at R = 2 kpc for all bar-forming models in (a) Group 1, (b) Group 2, (c) Group 3, and (d) Group 4. In (c) and (d), the shaded vertical bands indicate the duratio…
Figure 14
Figure 14. Figure 14: Variations of S/κ2 0 and |ξ| as functions of di￾mensionless time τ = tκ0 for QT = 1.2 and X = 1.5. The shaded regions denote the period of most significant growth, during which S < 0. To quantify the magnitude of swing amplification, we define the amplification factor…
Figure 15
Figure 15. Figure 15: Amplification factor Γ as a function of X for disks with QT = 1.2 (black solid), QT = 1.5 (red dashed), and QT = 2.0 (blue dotted). 1 tend to have larger values of X than those in Group 4, despite also having lower Vmax [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]
Figure 17
Figure 17. Figure 17: Simulation outcomes in the tOP–ϵELN plane. The symbols have the same meanings as in [PITH_FULL_IMAGE:figures/full_fig_p017_17.png]
Figure 18
Figure 18. Figure 18: Bar formation time tbar against the disk mass fraction fdisk in our bar-forming models. The symbols have the same meanings as in [PITH_FULL_IMAGE:figures/full_fig_p018_18.png]
Figure 19
Figure 19. Figure 19: Dependence on stellar mass M∗ of (a) bar strength A2/A0 and (b) bar length Rbar. Green dots and er￾ror bars represent the mean values and standard deviations from our bar-forming models. Bars within each group have identical stellar masses; data points are slightly of…
Figure 20
Figure 20. Figure 20: Relationship between the corotation radius RCR and the bar length Rbar for all bar-forming models in our simulations. The dashed and dotted lines correspond to R ≡ RCR/Rbar = 1.4 and 1.0, respectively. All bars satisfy R > 1.4, indicating that they are classified as s…

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